Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,24)(16,29)(17,23)(18,20)(19,34)(21,28)(22,32)(25,41)(26,42)(27,35)(30,40)(31,39)(33,38)(36,44)(37,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(14,20)(15,25)(22,29)(23,34)(26,30)(27,28)(38,39)(43,44);(2,4)(13,15,19,28,18,25,17,27)(14,21,23,24,20,35,34,41)(16,30,31,44,32,26,33,43)(22,37,39,42,29,36,38,40);(6,10)(9,11)(13,16)(14,22)(15,26)(17,33)(18,32)(19,31)(20,29)(21,36)(23,39)(24,40)(25,30)(27,44)(28,43)(34,38)(35,37)(41,42);(2,4)(7,12)(9,11)(13,17,18,19)(14,23,20,34)(15,27,25,28)(16,31,32,33)(21,24,35,41)(22,38,29,39)(26,43,30,44)(36,42,37,40)Polycyclic presentation:
2^9|b^2=fi,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590264);